On the Kummer construction (Q2269968)

The paper under review deals with a generalization of the construction of the Kummer surfaces. Namely, if \(A\) is an abelian surface ad \(i\) is an involution on \(A\), the Kummer surface \(K(A)\) is obtained via a two-steps construction: first one considers the quotient \(A/i\), then one resolves the singularities of \(A/i\). The generalization is the following: consider a \(d-\)dimensional abelian variety \(A\), an integer \(r\in\mathbb{N}\), and a finite group \(G\) with an irreducible integral representation \(\rho_{\mathbb{Z}}:G\longrightarrow GL(r,\mathbb{Z})\) with \(\{0\}\) as the fixed point set. If \(d\) is odd, assume that \(\text{det}(\rho_{\mathbb{Z}})=1\). The induced action is \(\rho_{A}:G\longrightarrow \text{Aut}(A^{r})\), where \(\rho_{A}:=\rho_{\mathbb{Z}}\otimes_{\mathbb{Z}}A\), i. e. \(G\) acts on \(A^{r}\) with integral matrices coming from \(\rho_{\mathbb{Z}}\), and we have the quotient \(Y:=A^{r}/G\). If a crepant resolution \(f:X\longrightarrow Y\) of \(Y\) exists, then \(X\) is a \(dr-\)dimensional manifold with trivial canonical divisor and \(H^{1}(X,\mathbb{C})=0\). The main issue of the paper is to give a method to calculate the Betti numbers of \(X\), which is carried out explicitely in several cases. In Section 2 of the paper, the author introduce notations and basic facts about groups, representations, crepant and symplectic resolutions. In Section 3 the general method for computing the cohomology of the crepant resolution \(X\) is described. Namely, the first step is to provide a description of the cohomology of \(Y=A^{r}/G\): if \(R(G)\) is the ring of complex representations of \(G\), \(\mu_{0}:R(G)\longrightarrow\mathbb{Z}\) the map assigning to a representation the rank of its maximal trivial subrepresentation, \(P_{Z}(t)=\sum_{i=0}^{\text{dim}(Z)}b_{i}(Z)t^{i}\) the virtual Poincaré polynomial of a complex algebraic variety, and \(P_{Z,G}(t)\) the \(G-\)Poincaré polynomial of \(Z\) with an action of a finite group \(G\), then \(P_{Y}(t)=\mu_{0}(P_{A^{r},G}(t))\) (see Lemma 3.1). The next step is to compute the cohomology of the crepant resolution \(f:X\longrightarrow Y\): the idea is to decompose \(Y\) in strata \(Y([H])\) for \(H\) subgroup of \(G\), where \(Y([H])\) is given by the points of \(Y\) whose isotropy group is conjugated to \(H\). Over \(Y([H])\) the singularities are all locally quotients of the form \(\mathbb{C}^{\text{dr}}/H\). This gives a decomposition of \(X\) in strata \(X([H])\) such that the map \(f([H]):X([H])\longrightarrow Y([H])\) is a locally trivial fiber bundle whose fiber depends on the resolution of \(\mathbb{C}^{\text{dr}}/H\). One computes \(P_{X}(t)\) via the \(P_{X([H])}(t)\). The authors make further assumption on \(f:X\longrightarrow Y\): it has to be a locally product (see Definition 3.2), and the McKay correspondence holds for crepant resolutions of \(\mathbb{C}^{\text{dr}}/H\). Principle 3.4 gives a general formula to compute \(P_{X}(t)\): it depends on \(G\), \(\rho_{\mathbb{C}}\) and on the integral conjugacy class of \(\rho_{\mathbb{Z}}\). In particular, constructions with the same \(\rho_{\mathbb{C}}\) but different \(\rho_{\mathbb{Z}}\) can have different cohomologies. In Section 4 the authors present explicit calculations for \(d=1\), i. e. \(A\) is an elliptic curve. For \(r=2\), then \(G=\mathbb{Z}_{n}\), the cyclic group of order \(n=2,3,4,6\). The case \(n=6\) is carried out. For \(r=3\) one gets a construction of a Calabi-Yau manifold via quotients. Then \(G\) can be the dihedral group \(D_{2a}\) for \(a=2,3,4,6\), the alternating group \(A_{4}\) and the symmetric group \(S_{4}\). Explicit computations are provided for \(S_{4}\) (see Proposition 4.3). In Section 5 the case \(d=2\), i. e. \(A\) is an abelian surface, is considered. The group is \(G=S_{r}\): in this case one gets \(X=\text{Kum}^{(r-1)}\), the generalized Kummer variety of dimension \(2r\). Explicit computation of \(P_{X}(t)\) is given for \(r=3\) (Proposition 5.2), and the general elements of the construction are presented in Lemma 5.1. In Section 6 the case of \(d=4\) is studied. The action of \(G\) is prescribed only on \(H^{1}(A,\mathbb{C})\), i. e. one fixes the complex representation \(\rho_{\mathbb{C}}:G\longrightarrow SL(\mathfrak{a})\) (where \(\mathfrak{a}\) is the tangent space of \(A\) at some point) without requiring that it comes from an integral representation. One has three cases: \(S_{r+1}\), \(G_{n,m}:=\mathbb{Z}^{n}_{m}\rtimes S_{n}\) and \(Q_{8}\rtimes\mathbb{Z}_{3}\). In the first two cases, the symplectic resolution is obtained via Hilbert schemes. In Theorem 6.1 the authors show that if \(G=Q_{8}\rtimes\mathbb{Z}_{3}\) acts with action on \(H^{1}(A,\mathbb{C})\) prescribed, then the quotient \(A/G\) does not admit any symplectic resolution: the proof is based on a careful study of the fixed locus of \(G\) and of its non-trivial subgroups. Theorems 6.2 and 6.4 present explicit formulas for \(P_{X}(t)\) for a crepant resolution \(X\) of \(A/G\), where \(G\) is respectively \(S_{3}\), the dihedral group of order 6, or \(\mathbb{Z}^{2}_{2}\rtimes\mathbb{Z}_{2}\), the dihedral group of order 8, with prescribed actions on \(H^{1}(A,\mathbb{C})\). In particular, the authors show that if two different crepant resolutions exists, then they have the same Betti numbers.